In this paper a new empirical failure criterion is proposed to predict rock strength under polyaxial stress conditions. The proposed strength criterion is a three-dimensional extension of the popular two-dimensional criterion by Bieniawski (1974). The criterion has two empirical constants that can be determined using standard triaxial compression tests. Using these same empirical constants, failure under polyaxial conditions can be predicted. The accuracy of the failure criterion has been established by conducting a series of triaxial compression and hollow cylinder tests on Apache Leap tuff. First, the results of the triaxial compression tests are used to determine the two empirical constants for Apache Leap tuff. Secondly, using these two constants, the failure of the hollow cylinders of Apache Leap tuff under various conditions are predicted and compared with the actual experimental results under the same conditions. The new empirical failure criterion matches the experimental results extremely well. Rock Mechanics Daemen & Schultz (eds).